Overlapping and communication policies in product development

Shorter product life cycles in many industries impel firms to accelerate the product development process. Overlapping development stages, combined with frequent information exchange, is commonly regarded as a core technique for faster product development. However, overlapping and communication require additional resources and can be costly. We investigate the time-cost tradeoffs involved in concurrent product development to determine the optimal overlapping and communication strategies. The methodology was applied to a refrigerator development process in order to illustrate its utility.     

Modular product design and product support

A modular design for a product or system can be an effective approach to support of the product if total replacement and repair times can be reduced by modular replacement. On the other hand, modular designs may involve higher costs of replacement and inventory holding. Cost models are developed for the evaluation of a modular design, and the distinction between integrated modules and replaceable subassemblies is discussed. Numerical examples are given to illustrate the tradeoffs to be made.     

Generalized hamiltonian circuits in the cartesian product of two-directed cycles

We find necessary and sufficient conditions for the existence of a closed walk that traverses r vertices twice and the rest once in the Cayley digraph of Z_m ⊕ Z_n . This is a generalization of the results known for r = 0 or 1.     

Disjoint circuits in the cartesian product of two directed cycles

We show that the Cartesian product of two directed cycles Z _a X Z _b has r disjointly embedded circuits C₁, C₂, ?, C_r with specified knot classes knot(C_i) = (m_i, n_i), for i = 1, 2, ?, r, if and only if there exist relatively prime non-negative integers m and n such that knot(C_i) = (m, n), for i = 1, 2, ?, r, and r(am + bn) ≦ ab. We generalize this result to the Cayley digraph on a finite abelian group with a two-element generating set.     

Factorizations of product graphs into cycles of uniform length

The following question is raised by Alspach, Bermond and Sotteau: IfG ₁ has a decomposition into hamilton cycles and a 1-factor, andG ₂ has a hamilton cycle decmposition (HCD), does their wreath productG ₁ *G ₂ admit a hamilton cycle decomposition? In this paper the above question is answered with an additional condition onG ₁. Further it is shown that some product graphs can be decomposed into cycles of uniform length, that is, the edge sets of the graphs can be partitioned into cycles of lengthk, for some suitablek.     

When the cartesian product of directed cycles is Hamiltonian

The cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the cartesian product C_n1 × C_n2 of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n₁ and n₂ is at least two and there exist positive integers d₁, d₂ so that d₁ + d₂ = d and g.c.d. (n₁, d₁) = g.c.d. (n₂, d₂) = 1. We also discuss some number-theoretic problems motivated by this result.     

Chromatic numbers of the strong product of odd cycles

The problem of determining the chromatic numbers of the strong product of cycles is considered. A construction is given proving χ (G) = 2^p +1 for a product of p odd cycles of lengths at least 2^p +1. Several consequences are discussed. In particular, it is proved that the strong product of p factors has chromatic number at most 2^p +1 provided that each factor admits a homomorphism to sufficiently long odd cycle C_mi, m_i ≥ 2^p +1.     

Global defensive alliances of trees and Cartesian product of paths and cycles

Given a graph $G$, a defensive alliance of $G$ is a set of vertices $S?V\left(G\right)$ satisfying the condition that for each $v\in S$, at least half of the vertices in the closed neighborhood of $v$ are in $S$. A defensive alliance $S$ is called global if every vertex in $V\left(G\right)?S$ is adjacent to at least one member of the defensive alliance $S$. The global defensive alliance number of $G$, denoted ${\gamma }_a\left(G\right)$, is the minimum size around all the global defensive alliances of $G$. In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete $k$-ary trees for $k=2,3,4$. We also establish upper bounds and lower bounds for ${\gamma }_a(P_m\times P_n),{\gamma }_a(C_m\times P_n)$ and ${\gamma }_a(C_m\times C_n)$, and show that the bounds are sharp for certain $m,n$.     

Presentations of the amalgamated free product of two infinite cycles

LetH=〈a,b;a ^k =b ^l 〉, wherek,l≧2 andk+l>4. McCool and Pietrowski have proved that any pair of generators forH is Nielsen equivalent to a pairx=a ^r andy=b ^s where $$(a){\text{ }}gcd(r, s) = gcd(r, k) = gcd(s, l) = 1,$$ $$(b){\text{ }}0< 2r \leqq ks{\text{ }}and{\text{ }}0< 2s \leqq lr.$$ In terms ofx andy,H can be presented as $$G = \left\langle {x,{\text{ }}y;{\text{ }}x^{ks} = y^{lr} ,\left[ {x,{\text{ }}y^l } \right] = \left[ {x^k ,{\text{ }}y} \right] = 1} \right\rangle$$ and Zieschang has shown that ifr=1 ors=1, thenH can be defined by a single relation inx andy. We establish the exact converse of Zieschang's result, namely thatH is not defined by a single relation inx andy unlessr=1 ors=1. The proof is based on an observation of Magnus which associates polynomials with relators and some elementary facts about cyclotomic polynomials.     

Factors affecting international product design

In this paper we examine the impact of changes of such factors as tariffs/import cost, exchange rate, and unit savings derived from economies of scale, on the product design of four international strategies which are characterised by two dimensions. The first dimension describes whether the company offers a standardised or a customised product. The second indicates whether the company centralises its production to a single facility in one country or decentralises its production to facilities located in each country. To address the above issue, we present a model that has elements from marketing and manufacturing.For the case where the product has one attribute, we show that when tariffs/import cost decrease, an international enterprise should respond by enhancing the features of its products. Similarly, the product features should be enhanced when the exchange rate increases or the unit savings derived from economies of scale increases. Numerical examples indicate that an international enterprise should change its production configuration from decentralised to centralised, in environments of high tariffs/import cost. Furthermore, an international enterprise should change its product policies from customised to standardised when the savings derived from economies of scale are high, and the exchange rate increases.     

When the cartesian product of two directed cycles is hyperhamiltonian

We say a digraph G is hyperhamiltonian if there is a spanning closed walk in G which passes through one vertex exactly twice and all others exactly once. We show the cartesian product Z_a × Z_b of two directed cycles is hyperhamiltonian if and only if there are positive integers m and n with ma + nb = ab + 1 and gcd(m, n) = 1 or 2. We obtain a similar result for the vertex-deleted subdigraphs of Z_a × Z_b.     

Structuring product development processes

This paper proposes operational frameworks for structuring product development processes. The primary objective of this research is to develop procedures to minimize iterations during the development process which adversely affect development time and costs. Several procedures are introduced to restructure the development process. The computation of the corresponding product development times is facilitated by two Markov models addressing different types of learning. The methodologies are employed to identify a set of managerial concerns in restructuring the product development processes.The developed framework has become an integral part of a re-engineering project for the development of rocket engines at Rocketdyne Division of Rockwell International. Throughout the paper, the methodologies are illustrated with the help of this process.     

When the cartesian product of two directed cycles is hypo-Hamiltonian

We show that the Cartesian product Z_a × Z_b of two directed cycles is hypo-Hamiltonian (Hamiltonian) if and only if there is a pair of relatively prime positive integers m and n with ma + nb = ab - 1 (ma + nb = ab). The result for hypo-Hamiltonian is new; that for Hamiltonian is known. These are special cases of the fact that there is a simple circuit of length p in Z_a × Z_b if and only if there is a pair of relatively prime non-negative integers m and n with ma + nb = p ≤ ab.     

Models for concurrent product and process design

We propose procedures to address product design and manufacturing process configurations concurrently in environments characterized by large degrees of product proliferation. Exploiting the intrinsic flexibility of product and process design, we present two approaches that synchronize production flows through the manufacturing system. These approaches integrate product and manufacturing system design decisions with operational concerns and provide powerful means for managing production in environments characterized by a proliferation of products. Experimental results show that the proposed methods can substantially reduce manufacturing lead times, work in process (WIP), and overall system complexity.     

Optimal overlapping and functional interaction in product development

Overlapping of development stages and interaction between different functions are regarded as important strategies for reducing development lead time. However, overlapping typically requires additional costs for rework and functional interaction increases communication time. This paper presents an analytical model to improve project performance by balancing the positive and negative effects of overlapping and functional interaction. We first investigate the progress of downstream development, which is essential to derive the optimal overlapping policies. We find that the downstream progress increases over time when the upstream evolution is fast or linear, but it is indefinite when the upstream evolution is slow. Then, we present optimal overlapping policies taking into account the complexity of downstream progress. The impact of different project properties, such as the dependency between development stages and the opportunity cost of time, on overlapping policies is discussed. Finally, we derive the optimal functional interaction strategy when the optimal overlapping is followed. The methodology is illustrated with a case study at a handset design company.     

Agent-based optimization for product family design

This paper presents a two-step approach to determine the optimal platform level for a selected set of product families and their variants. The first step employs a multi-objective optimization using an agent-based framework to determine the Pareto-design solutions for a given set of modules. The second step performs a post optimization analysis that includes application of the quality loss function (QLF) to determine the optimal platform level. The post optimization analysis yields the optimal platform level for a related set of product families and their variants. We demonstrate the working of the proposed method by using an example problem.